%I #14 Sep 15 2021 10:26:42

%S 1,1,4,1,13,52,1,40,130,520,2080,1,121,1210,4840,15730,62920,251680,1,

%T 364,11011,33880,44044,440440,1431430,1761760,5725720,22902880,

%U 91611520,1,1093,99463,925771,397852,12035023,37030840,120350230,48140092,481400920,1564552990

%N Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 3.

%C Abuse of notation: we write T(n, L) for T(n, k), where L is the k-th partition of n in A-St order.

%C For any permutation (e_1,...,e_r) of the parts of L, T(n, L) is the number of chains of subspaces 0 < V_1 < ··· < V_r = (F_3)^n with dimension increments (e_1,...,e_r).

%D R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.

%H Álvar Ibeas, <a href="/A347486/b347486.txt">First 20 rows, flattened</a>

%F T(n, (n)) = 1. T(n, L) = A022167(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.

%e The number of subspace chains 0 < V_1 < V_2 < (F_3)^3 is 52 = T(3, (1, 1, 1)). There are 13 = A022167(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 4 = A022167(2, 1) extensions to a two-dimensional subspace V_2.

%e Triangle begins:

%e k: 1 2 3 4 5 6 7

%e ----------------------------------

%e n=1: 1

%e n=2: 1 4

%e n=3: 1 13 52

%e n=4: 1 40 130 520 2080

%e n=5: 1 121 1210 4840 15730 62920 251680

%Y Cf. A036038 (q = 1), A022167, A015001 (last entry in each row).

%K nonn,tabf

%O 1,3

%A _Álvar Ibeas_, Sep 03 2021